Let and let . Then the Vandermonde matrix associated to is defined to be the matrix
For pairwise different (i. e. for ) matrix is invertible, as the following theorem proves:
Let be the Vandermonde matrix associated to the pairwise different points . Then the matrix whose -th entry is given by
is the inverse matrix of .
We prove that , where is the identity matrix.
Let . We first note that, by direct multiplication,
Therefore, if is the -th entry of the matrix , then by the definition of matrix multiplication
The Malgrange-Ehrenpreis theorem
Let be pairwise different. The solution to the equation
is given by
- , .
We multiply both sides of the equation by on the left, where is as in theorem 10.2, and since is the inverse of
we end up with the equation
Calculating the last expression directly leads to the desired formula.